Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces

Transcription

1 Plasmonics Plasmon: Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam

2 Flavours M-I-M Taper and wire Hybrid V-groove Wedge/hybrid

3 Dispersion Dispersion relation with loss - very large k, flat dispersion - also note superluminal part

4 1) Kramers-Kronig bounds on e 2) Back to the surface plasmon 3) Pulse propagation and dispersion

5 Kramers-Kronig Can a material have arbitrary real and imaginary e? No: material response functions are constrained by causality Frequency domain Time domain Note how convolution turns to product Physics: no response before cause

6 Kramers Kronig Either you have non-dispersive vacuum e=1, i.e., c=0, or - A window of real c implies a window of absorption - Real c(w) >0 means c(w) < 0 at other w (to avoid gain) - No dispersion but a refractive index not 1 is impossible Considerations hold for any physical response function

7 Typical solids Absorption bands close to intrinsic resonances Real n to the red also outside absorption Most materials have normal dispersion, i.e., goes up with energy is higher towards the blue is higher towards short Until you go through an absorption resonance

8 Dispersion relation in a Lorentzian gas Near resonance (g ~ 0.05 w0) - Strongly dispersive - Very low v g - Superluminal phase velocity - Negative & superluminal v g

9 Phase velocity A front of constant phase follows Phase velocity Suppose now I have a wave packet e.g. Note that at x=0, this is just a pulse

10 Group velocity Maybe you have dispersion The pulse envelope has a velocity known as the group velocity

11 Indices Group index [blue curve] very different from phase index [orange] Negative v g Superluminal v g, v f Region of strong absorption Kramers-Kronig in action

12 Movie dispersive propagation Strongly dispersive, weakly absorbing w c ~ 0.85 w 0 (here loss length > 50l) Pulse envelope tracks v g (Note phase fronts walk faster)

13 Strongly dispersive, weakly absorbing (here loss length > 50l) Pulse envelope tracks v g

14 Superluminal velocities Strongly dispersive, strongly absorbing w c ~ 1.03 w 0 v f =1.25c, vgroup=1.7c (at carrier frequency) Note how the packet - barely moves - v g >c not apparent - package break up Loss length ~ 3 l

15 Slow group v, superluminal phase v Superluminal phase Slow group velocity Weak abosrption w c ~ 1.1 w 0 v f =1.15c, vgroup=c/2 (at carrier frequency) Note how the group velocity has regained meaning Loss length ~ 20 l

16 Take home messages Phase velocity describes phase front propagation You get it from the ratio of w/c and k For weak absorption, group velocity describes Gaussian envelope Strong dispersion also entails strong absorption Superluminal phase and group velocities are common These are irrelevant to describe pulse-energy propagation Pulse break up (group velocity dispersion) Strong attenuation Plasmonics is strongly dispersive and at superluminality paradox Confinement, dispersion and absorption are all linked

17 Small scatterers and plasmonics

18 From plasmon to plasmonics k Plasmons in the bulk oscillate at w p determined by the free electron density and effective mass Plasmons confined to surfaces that can interact with light to form propagating surface plasmon polaritons (SPP) w drude drude p surface Ne me Ne m Confinement effects result in resonant SPP modes in nanoparticles w drude particle Ne me 0

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20 Observables Extinction cross section [m 2 ] Power removed from beam Incident intensity Extinction = scattering + absorption removed from the beam Re-radiated into all angles Lost as heat in the scatterer

21 Physical quantity - polarizability Small object kd <<1 - incident field is approximately constant Volume polarization (weak index so E=Ein) Total dipole moment Larger particles & e means - modified dipole moment - also higher order multipoles

22 Electrostatic sphere First consider a sphere in a static field No charge in Laplace equation means: ( r a) ( r a) E 0 e m e a q r z Boundary conditions set by D ( e E) ( e ) ( r a), e e ( r a), lim E z 1 2 m 2 0 r r r

23 Solution r p r E r E a r E r E r E r E m m m m m m m e q q q e e e e q q e e e q e e e e q cos cos cos cos cos cos cos E 0 e m e z r q a Easy to verify with 4 2 m SI SI m m p E a e e e e e e Inside sphere: homogeneous field Outside sphere: background field plus field of a dipole with In the ball: Outside:

24 Metal sphere p E a e e 3 m 4 e 0 easy 0 with easy e 2em Drude model for a metal: Lorentzian `plasmon resonance 2 2 w p 3 w 0 e 1 means easy a 2 2 w( w ig ) w0 w iwg Resonance where e(w 0 ) = -2 e m Response scales with the volume V exceeds V by factor 5 to 10 Shape shift condition e = -2 e m

25 Observables Extinction cross section [m 2 ] Power removed from beam Incident intensity Extinction = scattering + absorption removed from the beam Re-radiated into all angles Lost as heat in the scatterer

26 Cross section plasmonics Extinction crosssect. (m 2 ) nm Ag particle in glass E 2 / E in Wavelength (nm) Large plasmon particles (size > 50 nm diameter) s and at upper bound: unitary limit s 3 l 2 Plasmon particle is a solid state `strongest point-scatterer 2

27 Revisiting polarizability Classical model of harmonically bound electron describes atom, and scatterer alike as an oscillating dipole 2 p e03v w0 iwt iwt ( t) e ( ) 2 2 SI w e w0 w iwg E E Lorentzian resonance Extinction: how much power is taken from the beam? Cycle average work done by E on p dp W E Im dt in

28 Revisiting polarizability Extinction: how much power is taken from the beam (in SI units)? T iwt 1 iwt dpe 1 iwt iwt W Re[ Ee ] Re[ ] dt Re[ Ee ] Re[ iwee ] dt T dt T 0 0 T 1 iwt * iwt iwt * * iwt W ( e e ) ( iw e iw e ) dt 4T E E E E 0 Only cross terms survive cycle average T 1 * 2 2 W ( iw iw ) oscill.terms ( 2 w) dt 4T E E 0 W w Im E 2 2 T

29 Revisiting polarizability Classical model of harmonically bound electron Describes atom, and scatterer alike 2 p e03v w0 iwt iwt ( t) e ( ) 2 2 SI w e w0 w iwg E E Lorentzian resonance Scattering: how much power does p radiate? W SndA d r sinq Edipole d sphere r 2 psinq sinq 4e r 0 2 2

30 Optical theorem Equate extinction to scattering (energy conservation) Extinction Scattering m 2 4 k Im [ m ] Easy Work done to drive p Rayleigh / Larmor 8 k [ ] 3 Easy 1. Very small particles scatter like r 6 /l 4 (Rayleigh) 2. For very small particles absorption wins ~ r 3 /l 3. Big 2 implies large Im

31 Optical theorem Equate extinction to scattering (energy conservation) Extinction Scattering m 2 4 k Im [ m ] Easy Work done to drive p Rayleigh / Larmor 8 k [ ] 3 Easy 3 l Since Im Easy 22 Upper bound on the strongest possible dipole scatterer 3

32 Example: simple spheres Calculated exact cross section of Au spheres r=10-50 nm Dashed line: s = 4 k Im easy Surprises -Peak bounded by

33 Revisiting polarizability Classical model of harmonically bound electron describes atom, and scatterer alike as an oscillating dipole 2 p e03v w0 iwt iwt ( t) e ( ) 2 2 SI w e w0 w iwg E E Lorentzian resonance Compared to electrostatics: g must be adapted to contain radiation damping

34 Summary Small particles of size < l/10 scatter like dipoles Generally: 3x3 polarizability tensor proportional to V Depolarization factors require microscopic model General features of polarizability Optical theorem contrains Polarizability bounded by above by unitary limit Radiative damping broadens resonances & raises albedo

35 Photochemical imaging Plasmon particles in azo-polymer Regions of high field contract Measured using atomic force microscopy Wiederrecht

36 Uses of particle plasmons Very strong local fields E 2 : 10 4 x stronger than incident Au spheres 5,8,20 nm, gaps of 1-3 nm E 2 / E in times enhanced optical field intensity

37 Resonance shifts Single protein binding & unbinding

38 Plasmon ruler Idea: Alivisatos group

39 Single molecules [Moerner & Orrit, 89] Keep on diluting molecules 100 micron 1 molecule can emit about 10 7 photons per second (1 pw) Observable with a standard [6k ] CCD camera + NA=1.4 objective

40 Ultimate control over light micrometers High Q nanometers Ultrasmall V Interference-based Material-based free-electrons

41 Patch antenna 1000x more light per molecule

42 Gain 1: funnel more pump photons onto emitter Gain 2: faster emission after excitation Gain 3: beaming of emission into narrower angle range

43 Count rates Single quantum dot Almost 2000x more photons / s at the same input flux

44 Funneling light into a single beam Sample: perforated Au film - hexagons of 440 nm pitch Sources: dilute fluorophores Atto 640 dye diffusing in H 2 O We pump the central hole only in a confocal microscope L. Langguth, D. Punj, J. Wenger, A. F. Koenderink ACS Nano 7, 8840 (2013)

45 Emission can be redirected k y Single hole One shell Two shells Three shells Single molecule phased array antenna Fourier image k x (up to NA=1.2) Emission strongly redirected in a narrow beam Single aperture: 1. Emission 10x brightness launched enhancement into surface (full wave NA), pump E 2 Array: 2. Diffraction 40x enhancement at normal in incidence forward direction pitch d=l SPP L. Langguth, D. Punj, J. Wenger, A. F. Koenderink ACS Nano 7, 8840 (2013)

46 Directional scattering, phased arrays Arrays of particles can result in directional scattering Arrays of particles can result in directional emission When a local source is embedded

47 Phased array Suppose I have N dipoles, radiating into the far field as Taylor expand for large Viewing distance r 1 r 2

48 Far field If all dipoles are parallel (not meaning in phase) Overall spherical wave Array geometry Structure factor Single scatterer Form factor

49 Yagi-Uda Optical domain: a chain of plasmon particles driven by a single molecule

50 Yagi Uda antenna Curto et al., Science (2010).

51 Phased array emission control Diffractive arrays for LEDs Lozano, Light Sci Appl (2013) Torma & Barnes, Rep Prog Phys (2015)

52 Uses of particle plasmonics Enhanced EM fields for spectroscopy Sensing single molecules 2000-fold enhanced fluorescence brightness Enhanced capture of sunlight in solar cells Enhanced super-small photodetectors Enhanced photo-chemistry Quantum optics

53 Exercise-help - particle dimer

54 Example dimer in static approximation Dimer in a static approximation Linear problem Symmetric, but not real matrix 1/polarizability on the diagonal Interaction on the off-diagonal - this will shift resonances

55 Hybridization (exercise)

56 Exercise help - Kramers Kronig

57 Causal response Example of an allowable response function in time What does Imply for

58 Even and odd / real, imaginary Three important realizations: 1) c(t) is a real function 2) It s even part contributes to the real part of c(w) 3) It s odd part contributes to the imaginary part of c(w)

59 So what is the even / odd part? Define as odd part of the response function This has no physical meaning, just a math trick

60 So what is the even / odd part? The even part of c(t) follows from the odd part So now: Fourier transform of a product is a convolution of FTs

61 So what is the even / odd part? So what is the Fourier transform of sgn(t)? Remember

62 Kramers-Kronig Therefore the imaginary part of the frequency-dependent Permittivity completely specifies the real part Similarly